Past exam of the biological physics course of the University of Oxford / 2015 / 3

Write $e=q>0$ for the elementary charge and $ϵ={ϵ}_0{ϵ}_r$ for the permittivity. Taking the bulk electric potential as zero, the Boltzmann distribution of point ions gives $c_i(x)=c_i^{\infty }\exp [-e{z}_i\varphi (x)/(k_{B}T)]$. Here $c_i$ is a number density, not a molar concentration. Combining this with Poisson's equation gives the planar Poisson-Boltzmann equationMultiplying by ${\varphi }^{\prime }$ and integrating from the neutral bulk, where $\varphi ={\varphi }^{\prime }=0$, yieldsFor the solution occupying $x>0$, with the field on the other side neglected, Gauss's law gives $\sigma =-ϵ{\varphi }^{\prime }(0)$. On the branch that decays to zero without crossing it,Source sign qualification: the unsigned square root printed in the paper gives $|\sigma |$. A negative wall electric potential, such as the next part's $-73\mathrm{mV}$, has ${\varphi }^{\prime }(0)>0$ and hence $\sigma <0$, providing a concrete counterexample to interpreting that printed expression as signed surface charge density. This is the planar Grahame equation, with an explicit sign convention.

Full dissociation gives bulk concentrations in $\mathrm{mM}$ of $155$ for sodium, $5$ for potassium, $2$ for calcium, $114$ for chloride, and $25$ for hydrogen phosphate. The positive and negative charge concentrations are both $164\mathrm{mM}$, so the bulk is electrically neutral. At $T=298.15\mathrm{K}$,The Boltzmann distribution then givesFor completeness the coion concentrations are $114e^{-\nu }=6.65196\mathrm{mM}$ and $25e^{-2\nu }=0.0851195\mathrm{mM}$. They are depleted, not zero. Using $1\mathrm{mM}=1\mathrm{mol}{\mathrm{m}}^{-3}$ and multiplying molar concentrations by the Avogadro constant in the Grahame equation givesThis calculation retains every species. Since $|{\varphi }_0|/V_T=2.84$, a linear Debye–Hückel approximation is not justified. The large accumulation, especially of divalent counterions, follows from the valence in the Boltzmann factor. However, multi-molar local concentrations also warn that point-ion mean-field theory neglects finite ion size, hydration, specific adsorption, correlations, and changes in local permittivity. These are predictions of the stated idealized model, not precise measured interfacial concentrations.

Assume the structural surface charge density stays fixed while a divalent chloride is added, without changing $T$ or $ϵ$. Let $v=e{\varphi }_0/(k_{B}T)$ and $\delta c$ be the increment of the bulk divalent concentration. Electroneutral addition also increases bulk chloride by $2\delta c$. Varying the squared Grahame equation at fixed $\sigma$ givesAt a strongly negative wall, coions are small. Dropping their contribution to the denominator and $2e^v$ in the numerator gives the requested leading approximationThe $-3$ includes the change in bulk osmotic concentration of one added divalent counterion and its two monovalent coions. The numerator's increment is a bulk concentration; the denominator contains surface concentrations.^[ref] Extra counterions screen the wall and make its negative electric potential less negative. At fixed bulk calcium, $c_{\mathrm{Ca}}(0)=c_{\mathrm{Ca}}^{\infty }{e}^{-2{\varphi }_0/V_T}$ therefore decreases. Under the stipulated proportionality, synaptic activity decreases as well; this argument isolates electrostatic accumulation rather than assuming a separate specific magnesium effect on the ion channels.

For $\delta c=5\mathrm{mM}$, the printed linear approximation gives $\delta {\varphi }_0=9.53434\mathrm{mV}$. Linearizing the activity too gives $\delta A/A=-2\delta {\varphi }_0/V_T=-0.7422$, or approximately a $74\%$ reduction. Exponentiating this approximate voltage instead gives $A^{\prime }/A=0.4761$. Neither is a controlled finite-increment prediction: the added divalent concentration is 2.5 times its initial value.

A consistent finite-change calculation is possible within the same Poisson-Boltzmann equation. Define, using $\mathrm{mM}$ throughout,Fixed $\sigma$ requires $S_5(u^{\prime })=S_0(2.84129)$. The physical positive root is $u^{\prime }=2.58241$, givingThus the self-consistent finite-change estimate is about a 40.4% reduction. Both estimates rely on fixed surface charge, ideal ionic activities, and unchanged bulk calcium; the limitations of the large local concentrations remain.